A radioactive isotope has a decay constant of 9.68E-03 /s. Calculate the time taken for three quarters of a sample of the isotope to decay.
The decay constant, denoted by λ, is given as 9.68E-03 /s.
The decay of a radioactive sample can be modeled using the equation: N(t) = N0 * e^(-λt),
where N(t) is the number of radioactive atoms remaining at time t, N0 is the initial number of radioactive atoms, λ is the decay constant, and e is the base of the natural logarithm.
We need to calculate the time taken for three-quarters (3/4) of the sample to decay, which means we want to find t when N(t) = 1/4 * N0.
1/4 * N0 = N0 * e^(-λt).
Dividing both sides of the equation by N0, we get:
1/4 = e^(-λt).
To isolate t, take the natural logarithm (ln) of both sides:
ln(1/4) = ln(e^(-λt)).
Using the property of logarithms, ln(a^b) = b * ln(a), we have:
ln(1/4) = -λt * ln(e).
Since ln(e) is equal to 1, we can simplify the equation further:
ln(1/4) = -λt.
Now, we solve for t:
-λt = ln(1/4).
Dividing both sides by -λ, we get:
t = ln(1/4) / (-λ).
Plugging in the given decay constant, λ = 9.68E-03 /s, we can now calculate t:
t = ln(1/4) / (-9.68E-03 /s).
Using a calculator, evaluate ln(1/4) / (-9.68E-03) to get the time taken for three-quarters of the sample to decay.
list of prime numbers from 50 through 75
The prime numbers from 50 through 75 are:
53, 59, 61, 67, 71, 73
To calculate the time taken for three quarters (75%) of a radioactive isotope sample to decay, we can use the formula for exponential decay:
N(t) = N₀ * e^(-λt)
where:
N(t) is the remaining amount of the isotope at time t
N₀ is the initial amount of the isotope
λ is the decay constant
t is the time
In this case, we want to solve for t when N(t) = 0.75N₀ (three quarters of the initial amount).
So, the equation becomes:
0.75N₀ = N₀ * e^(-λt)
Dividing both sides by N₀:
0.75 = e^(-λt)
To isolate t, take the natural logarithm (ln) of both sides:
ln(0.75) = -λt
Now we can solve for t:
t = (ln(0.75)) / -λ
Using the given decay constant, λ = 9.68E-03 /s:
t = ln(0.75) / (-9.68E-03 /s)
Calculating this value:
t ≈ 28.77 seconds
Therefore, it would take approximately 28.77 seconds for three quarters of the sample of the isotope to decay.